Gradient and transport estimates for the heat flow on non-convex domains

For the Neumann heat flow on nonconvex Riemannian domains $D\subset M$, we provide sharp gradient estimates and transport estimates with a novel $\sqrt{t}$-dependence, for instance, $$\text{Lip}(P_t^{D}f)\le e^{2S\sqrt{t/\pi }+\mathcal{O}(t)}\cdot \text{Lip}(f),$$ and we provide an equivalent characterization of the lower bound $S$ on the second fundamental form of the boundary in terms of these quantitative estimates.